Oxford Mathematician Ben Green on how and why he has been pondering footballs in high dimensions.

"A 3-dimensional football is usually a truncated icosahedron. This solid has the virtue of being pleasingly round, hence its widespread use as a football. It is also symmetric in the sense that there is no way to tell two different vertices apart: more mathematically, there is a group of isometries of $\mathbf{R}^3$ acting transitively on the vertices.

The Coronavirus disease pandemic (COVID-19) poses unprecedented challenges for governments and societies around the world. In addition to medical measures, non-pharmaceutical measures have proven to be critical for delaying and containing the spread of the virus. However, effective and rapid decision-making during all stages of the pandemic requires reliable and timely data not only about infections, but also about human behaviour, especially on mobility and physical co-presence of people. 

Executive stock options (ESOs) are contracts awarded to employees of companies, which confer the right to reap the profit from buying the company stock (exercising the ESO) at or before a fixed maturity time $T$, for a fixed price specified in the contract (the strike price of the ESO). ESOs are used to augment the remuneration package of employees, the idea being to give them an incentive to boost the company's fortunes, and thus the stock price, making their ESO more valuable.

Social distancing measures to reduce the spread of the novel coronavirus are in place worldwide. These guideline are for everyone. We are all expected to reduce our contact with others, and this will have some negative impacts in terms of mental health and loneliness, particularly for the elderly and other vulnerable groups. So why should we follow measures that seem so extreme? The answer is simple. Social distancing works. It reduces transmission of the virus effectively and lessens the impact on already stretched healthcare services.

Several well-known formulas involving reflection groups of finite-dimensional algebraic systems break down in infinite dimensions, but there is often a predictable way to correct them. Oxford Mathematician Thomas Oliver talks about his research getting to grips with what structures underlie the mysterious correction process.

Oxford Mathematician Weijun Xu talks about his exploration of the universal behaviour of large random systems:

"Nature comes with a separation of scales. Systems that have apparently different individual interactions often behave very similarly when looked at from far away. This phenomenon is particularly attractive when randomness is involved.

Mathematics has long played a crucial role in understanding ecological dynamics in a range of ecosystems, from classical models of predators and prey to more recent models of aquatic organisms' interaction with global climate. When we think of ecosystems we usually imagine coral reefs or tropical forests; however, over the past decade a substantial effort has emerged in studying a tiny, yet deadly ecosystem: human tumours. Rather than being a single malignant mass, tumours are living and evolving ecosystems.

Networks provide powerful tools and methods to understand inter-connected systems. What is the most central element in a system? How does its structure affect a linear or non-linear dynamical system, for instance synchronisation? Is it possible to find groups of elements that play the same role or are densely connected with each other? These are the types of questions that can be answered within the realm of network science. For standard algorithms to be used, however, it is usually expected that the underlying network structure is known.

In 2018, the World Health Organization added “Disease X” to its list of priority diseases, alongside diseases like Ebola virus disease and SARS. Disease X is representative of infectious agents that are not currently known to cause cases in humans. In other words, it denotes the possibility of an epidemic of a disease that we have never seen before.

Oxford Mathematician Patrick Kidger talks about his recent work on applying the tools of controlled differential equations to machine learning.

Sequential Data

The changing air pressure at a particular location may be thought of as a sequence in $\mathbb{R}$; the motion of a pen on paper may be thought of as a sequence in $\mathbb{R}^2$; the changes within financial markets may be thought of as a sequence in $\mathbb{R}^d$, with $d$ potentially very large.